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Edexcel S1 Q2
11 marks Easy -1.3
2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.
Number of daisies\(1 \mid 1\) means 11
11223444(7)
15567899(7)
200133334(8)
25567999(7)
3001244(6)
366788(5)
413(2)
  1. Write down the modal value of these data.
  2. Find the median and the quartiles of these data.
  3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
  4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
  5. Comment on how the student might summarise both sets of raw data before drawing box plots.
    (1 mark)
Edexcel S1 Q3
11 marks Standard +0.3
3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by \(x\) and the corresponding frequency for that group by \(f\). The data were then coded using \(y = \frac { ( x - 755.0 ) } { 2.5 }\) and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$
  1. Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
    (9 marks)
    An estimate of the interquartile range for these data was 27.7 hours.
  2. Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
    (2 marks)
Edexcel S1 Q4
14 marks Standard +0.3
4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .
  1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
    (2 marks)
    The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
  2. Find the probability distribution of \(A\).
  3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
  4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
  5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).
Edexcel S1 Q5
15 marks Moderate -0.8
5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.
  1. Represent this information on a tree diagram.
  2. Find the probability that on any particular day he uses the sauna.
  3. Given that he uses the sauna one day, find the probability that he had been swimming.
  4. Given that he did not use the sauna one day, find the probability that he had been swimming.
Edexcel S1 Q6
16 marks Moderate -0.8
6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of \(x\), the test rig speed in miles per hour (mph), and the temperature, \(y ^ { \circ } \mathrm { C }\), generated in the shoulder of the tyre for a particular tyre material.
\(x ( \mathrm { mph } )\)1520253035404550
\(y \left( { } ^ { \circ } \mathrm { C } \right)\)53556365788391101
  1. Draw a scatter diagram to represent these data.
  2. Give a reason to support the fitting of a regression line of the form \(y = a + b x\) through these points.
  3. Find the values of \(a\) and \(b\).
    (You may use \(\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615\) )
  4. Give an interpretation for each of \(a\) and \(b\).
  5. Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table. A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
  6. Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.
Edexcel S1 2003 November Q1
16 marks Moderate -0.8
  1. A company wants to pay its employees according to their performance at work. The performance score \(x\) and the annual salary, \(y\) in \(\pounds 100\) s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.
\(x\)15402739271520301924
\(y\)216384234399226132175316187196
$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$
  1. Draw a scatter diagram to represent these data.
  2. Calculate exact values of \(S _ { x y }\) and \(S _ { x x }\).
    1. Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\). Give the values of \(a\) and \(b\) to 3 significant figures.
    2. Draw this line on your scatter diagram.
  4. Interpret the gradient of the regression line. The company decides to use this regression model to determine future salaries.
  5. Find the proposed annual salary for an employee who has a performance score of 35 .
Edexcel S1 2003 November Q2
18 marks Standard +0.3
2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.
  1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
  2. Find the probability distribution of \(X\).
  3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
  4. Find the probability that Linda scores more points in round 2 than in round 1.
Edexcel S1 2003 November Q3
9 marks Moderate -0.8
3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .
    1. Find the probability of a jar containing less than the stated weight.
    2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that \(1 \%\) of the jars contain less than the stated weight. The standard deviation stays the same.
  2. Find the new mean weight of sauce.
Edexcel S1 2003 November Q4
7 marks Easy -1.2
4. Explain what you understand by
  1. a sample space,
  2. an event. Two events \(A\) and \(B\) are independent, such that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
    Find
  3. \(\mathrm { P } ( A \cap B )\),
  4. \(\mathrm { P } ( A B )\),
  5. \(\mathrm { P } ( A \cup B )\).
Edexcel S1 2003 November Q5
9 marks Moderate -0.8
5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),
  1. show that \(n = 9\). Find
  2. \(\mathrm { P } ( X < 7 )\),
  3. \(\operatorname { Var } ( X )\).
Edexcel S1 2003 November Q6
16 marks Moderate -0.8
6. A travel agent sells holidays from his shop. The price, in \(\pounds\), of 15 holidays sold on a particular day are shown below.
29910502315999485
3501691015650830
992100689550475
For these data, find
  1. the mean and the standard deviation,
  2. the median and the inter-quartile range. An outlier is an observation that falls either more than \(1.5 \times\) (inter-quartile range) above the upper quartile or more than \(1.5 \times\) (inter-quartile range) below the lower quartile.
  3. Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, \(\pounds x\), of each of 20 holidays sold on the website. The cheapest holiday sold was \(\pounds 98\), the most expensive was \(\pounds 2400\) and the quartiles of these data were \(\pounds 305 , \pounds 1379\) and \(\pounds 1805\). There were no outliers.
  4. On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
  5. Compare and contrast sales from the shop and sales from the website. \section*{END}
Edexcel S1 2004 November Q1
14 marks Moderate -0.8
  1. As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.
TotalsAsifTotals
(9)87432110184457(4)
(11)9865433111957899(5)
(6)87422020022448(6)
(6)943100212356679(7)
(4)6411221124558(7)
(2)202311346678(8)
(2)71242489(4)
(1)9254(1)
(2)9326(0)
Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.
KeithAsif
Lower quartile191\(a\)
Median\(b\)218
Upper quartile221\(c\)
  1. Find the values of \(a , b\) and \(c\). Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
  2. On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
  3. Comment on the skewness of the two distributions.
Edexcel S1 2004 November Q2
4 marks Moderate -0.8
2. An experiment carried out by a student yielded pairs of \(( x , y )\) observations such that $$\bar { x } = 36 , \quad \bar { y } = 28.6 , \quad S _ { x x } = 4402 , \quad S _ { x y } = 3477.6$$
  1. Calculate the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). Give your values of \(a\) and \(b\) to 2 decimal places.
  2. Find the value of \(y\) when \(x = 45\).
Edexcel S1 2004 November Q3
12 marks Standard +0.3
3. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$
  1. In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
    1. Show that the value of \(\sigma\) is 5 .
    2. Find the value of \(\mu\).
  3. Find \(\mathrm { P } ( 69 \leq X \leq 83 )\).
Edexcel S1 2004 November Q4
14 marks Easy -1.3
4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2 \\ \alpha , & x = - 1,0 \\ 0.1 , & x = 1,2 . \end{array}$$ Find
  1. \(\alpha\),
  2. \(\mathrm { P } ( - 1 \leq X < 2 )\),
  3. \(\mathrm { F } ( 0.6 )\),
  4. the value of \(a\) such that \(\mathrm { E } ( a X + 3 ) = 1.2\),
  5. \(\operatorname { Var } ( X )\),
  6. \(\operatorname { Var } ( 3 X - 2 )\).
Edexcel S1 2004 November Q5
7 marks Easy -1.3
5. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }\).
  1. Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)
Edexcel S1 2004 November Q6
18 marks Easy -1.2
6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 , \\ \Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$
  1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
  2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
  4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
  5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
  6. Comment on Mr Brawn's assumption of normality.
Edexcel S1 2004 November Q7
6 marks Easy -1.8
7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.
TimeNumber of students
\(40 - 44\)10
\(45 - 47\)15
4823
\(49 - 51\)21
\(52 - 55\)16
\(56 - 60\)15
  1. Give a reason to support the use of a histogram to represent these data.
  2. Write down the upper class boundary and the lower class boundary of the class 40-44.
  3. On graph paper, draw a histogram to represent these data. END
Edexcel S2 2014 January Q1
8 marks Moderate -0.3
  1. The probability of a leaf cutting successfully taking root is 0.05
Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be
    1. exactly 1
    2. more than 2 A second random sample of 160 leaf cuttings is selected.
  2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
Edexcel S2 2014 January Q2
10 marks Moderate -0.3
2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.
  1. Suggest a suitable sampling frame for the sample survey.
  2. Identify the sampling units.
  3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
  4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.
Edexcel S2 2014 January Q3
11 marks Standard +0.3
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$
  1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
  2. Use calculus to find (i) \(\mathrm { E } ( X )\) (ii) \(\operatorname { Var } ( X )\).
Edexcel S2 2014 January Q4
7 marks Standard +0.3
  1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).
Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out. \(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)
  1. Write down the alternative hypothesis.
  2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
  3. find the largest possible value of \(\lambda\) that can be found by using the tables.
Edexcel S2 2014 January Q5
12 marks Standard +0.8
5. A school photocopier breaks down randomly at a rate of 15 times per year.
  1. Find the probability that there will be exactly 3 breakdowns in the next month.
  2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
  3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
  4. Find the probability that the head teacher will buy a new photocopier.
Edexcel S2 2014 January Q6
15 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find the median of \(X\), giving your answer to 3 significant figures.
Edexcel S2 2014 January Q7
12 marks Challenging +1.2
  1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).
Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).